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Volume densities with the mean value property for harmonic functions. (English) Zbl 0818.31003
Let $$U$$ be a bounded domain in $$\mathbb{R}^ d$$ $$(d\geq 2)$$ such that $$0\in U$$, and let $$\lambda$$ denote Lebesgue measure on $$U$$. It is shown that there exists a (strictly) positive $$C^ \infty$$ function $$w$$ on $$U$$ such that $$h(0)= \int hw d\lambda$$ for every $$h$$ in $${\mathcal H}_ b (U)$$, the set of bounded harmonic functions on $$U$$. Further, if $$\partial U$$ is of class $$C^{1+ \varepsilon}$$, then it can be arranged that $$\inf w(U)>0$$. The smoothness condition on $$\partial U$$ cannot be omitted: the authors construct a domain $$U$$ such that if $$w$$ is a nonnegative measurable function on $$U$$ which satisfies $$h(0)= \int hw d\lambda$$ for every $$h$$ in $${\mathcal H}_ b (U)$$, then $$\inf w(U)= 0$$. This answers a question posed by A. Cornea at the International Conference on Potential Theory in Nagoya in 1990.

##### MSC:
 31B05 Harmonic, subharmonic, superharmonic functions in higher dimensions 31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
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##### References:
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